Editing Stereographic projection (section)

===Generalizations===
More generally, stereographic projection may be applied to the unit [[n-sphere|{{math|''n''}}-sphere]] {{math|''S''<sup>''n''</sup>}} in ({{math|''n'' + 1}})-dimensional [[Euclidean space]] {{math|'''E'''<sup>''n''+1</sup>}}.  If {{math|''Q''}} is a point of {{math|''S''<sup>''n''</sup>}} and {{math|''E''}} a [[hyperplane]] in {{math|'''E'''<sup>''n''+1</sup>}}, then the stereographic projection of a point {{math|''P'' ∈ ''S''<sup>''n''</sup> − {{mset|''Q''}}}} is the point {{math|''{{prime|P}}''}} of intersection of the line {{math|{{overline|''QP''}}}} with {{math|''E''}}. In [[Cartesian coordinates]] ({{math|''x''{{sub|''i''}}}}, {{math|''i''}} from 0 to {{math|''n''}}) on {{math|''S''<sup>''n''</sup>}} and ({{math|''X''{{sub|''i''}}}}, {{math|''i''}} from 1 to ''n'') on {{math|''E''}}, the projection from {{math|1=''Q'' = (1, 0, 0, ..., 0) ∈ ''S''<sup>''n''</sup>}} is given by
<math display="block">X_i = \frac{x_i}{1 - x_0} \quad (i = 1, \dots, n).</math>
Defining
<math display="block">s^2=\sum_{j=1}^n X_j^2 = \frac{1+x_0}{1-x_0},</math>
the inverse is given by
<math display="block">x_0 = \frac{s^2-1}{s^2+1} \quad \text{and} \quad x_i = \frac{2 X_i}{s^2+1} \quad (i = 1, \dots, n).</math>

Still more generally, suppose that {{math|''S''}} is a (nonsingular) [[quadric|quadric hypersurface]] in the [[projective space]] {{math|'''P'''<sup>''n''+1</sup>}}. In other words, {{math|''S''}} is the locus of zeros of a non-singular quadratic form {{math|''f''(''x''<sub>0</sub>, ..., ''x''<sub>''n''+1</sub>)}} in the [[homogeneous coordinates]] {{math|''x''<sub>''i''</sub>}}. Fix any point {{math|''Q''}} on {{math|''S''}} and a hyperplane {{math|''E''}} in {{math|'''P'''<sup>''n''+1</sup>}} not containing {{math|''Q''}}.  Then the stereographic projection of a point {{math|''P''}} in {{math|''S'' − {{mset|''Q''}}}} is the unique point of intersection of {{math|{{overline|''QP''}}}} with {{math|''E''}}.  As before, the stereographic projection is conformal and invertible on a non-empty Zariski open set. The stereographic projection presents the quadric hypersurface as a [[rational variety|rational hypersurface]].<ref>Cf. Shafarevich (1995).</ref>  This construction plays a role in [[algebraic geometry]] and [[conformal geometry]].